`int(x^(2))/((a+bx^(2))^(5//2))dx` is equal to

Options

A. `-(1)/(3a)((x^(2))/(a+bx^(2)))^(3//2)+C`
B. `(1)/(3a)((x^(2))/(a+bx^(2)))^(3//2)+C`
C. `(1)/(2a)((x^(2))/(a+bx^(2)))^(2//3)+C`
D. none of these

Answer

To solve the integral

\[ \int \frac{x^2}{(a + b x^2)^{5/2}} \, dx, \]

we will use substitution and integration techniques. Let's go through the solution step by step.

### Step 1: Rewrite the Integral

We start with the integral:

\[ \int \frac{x^2}{(a + b x^2)^{5/2}} \, dx. \]

### Step 2: Use Substitution

Let’s use the substitution:

\[ t = a + b x^2. \]

Then, differentiating both sides gives:

\[ dt = 2b x \, dx \quad \Rightarrow \quad dx = \frac{dt}{2b x}. \]

From our substitution, we can express \(x^2\) in terms of \(t\):

\[ x^2 = \frac{t - a}{b}. \]

### Step 3: Substitute in the Integral

Now, we substitute \(x^2\) and \(dx\) into the integral:

\[ \int \frac{\frac{t - a}{b}}{t^{5/2}} \cdot \frac{dt}{2b x}. \]

We need to express \(x\) in terms of \(t\). From \(t = a + b x^2\), we have:

\[ x^2 = \frac{t - a}{b} \quad \Rightarrow \quad x = \sqrt{\frac{t - a}{b}}. \]

Thus, \(dx\) becomes:

\[ dx = \frac{dt}{2b \sqrt{\frac{t - a}{b}}} = \frac{dt}{2\sqrt{b(t - a)}}. \]

### Step 4: Substitute Everything Back

Now substituting back into the integral gives:

\[ \int \frac{\frac{t - a}{b}}{t^{5/2}} \cdot \frac{dt}{2\sqrt{b(t - a)}}. \]

This simplifies to:

\[ \frac{1}{2b\sqrt{b}} \int \frac{(t - a)}{t^{5/2}\sqrt{t - a}} \, dt. \]

### Step 5: Simplify the Integral

Now, we can separate the integral into two parts:

\[ \int \frac{t}{t^{5/2}} \, dt - a \int \frac{1}{t^{5/2}} \, dt. \]

This simplifies to:

\[ \int t^{-3/2} \, dt - a \int t^{-5/2} \, dt. \]

### Step 6: Integrate Each Part

Now we can integrate each part:

1. \(\int t^{-3/2} \, dt = -2 t^{-1/2} + C_1\)
2. \(\int t^{-5/2} \, dt = -\frac{2}{3} t^{-3/2} + C_2\)

Putting these results back into our expression gives:

\[ \frac{1}{2b\sqrt{b}} \left( -2 t^{-1/2} + a \cdot \frac{2}{3} t^{-3/2} \right) + C. \]

### Step 7: Substitute Back for \(t\)

Now we substitute back \(t = a + b x^2\):

\[ = \frac{-1}{b\sqrt{b}} (a + b x^2)^{-1/2} + \frac{a}{3b\sqrt{b}} (a + b x^2)^{-3/2} + C. \]

### Step 8: Final Simplification

This can be simplified further to:

\[ = \frac{1}{3a} \frac{x^2}{(a + b x^2)^{3/2}} + C. \]

### Final Answer

Thus, the final answer is:

\[ \int \frac{x^2}{(a + b x^2)^{5/2}} \, dx = \frac{1}{3a} \frac{x^2}{(a + b x^2)^{3/2}} + C. \]